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  • Open Access

Euler Buckling on Curved Surfaces

Shiheng Zhao (赵世恒) and Pierre A. Haas*

  • *Contact author: haas@pks.mpg.de
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Phys. Rev. Lett. 135, 247201 – Published 10 December, 2025

DOI: https://doi.org/10.1103/63py-ph5s

Abstract

Euler buckling epitomizes mechanical instabilities: an inextensible straight elastic line in the plane buckles under compression when the compressive force F reaches a critical value F*>0. But how does an elastic line buckle within a general curved surface? Here, we reveal that the classical instability changes fundamentally: by weakly nonlinear analysis of the buckling of an asymptotically short elastic line, we show that the critical force for the lowest buckling mode is F*=0 and discover a new bifurcation structure in which the modes of classical Euler buckling split into pairs. For long elastic lines, we numerically find an additional bifurcation by which the second of these new modes becomes the lowest mode and show that, at sufficiently large F, they snap discontinuously to higher end-to-end compression. Our results constitute the foundations for a class of buckling instabilities that arise within curved surfaces, for example when biological shape emerges in development.

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Outline

Information

Phys. Rev. Lett. 135, 247201– Published 10 December, 2025
  • Received 4 April 2025
  • Revised 31 July 2025
  • Accepted 19 November 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

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